Learning to Handle Complex Constraints for Vehicle Routing Problems

Vehicle Routing Problems (VRPs) can model many real-world scenarios and often involve complex constraints. While recent neural methods excel in constructing solutions based on feasibility masking, they struggle with handling complex constraints, especially when obtaining the masking itself is NP-hard. In this paper, we propose a novel Proactive Infeasibility Prevention (PIP) framework to advance the capabilities of neural methods towards more complex VRPs. Our PIP integrates the Lagrangian multiplier as a basis to enhance constraint awareness and introduces preventative infeasibility masking to proactively steer the solution construction process. Moreover, we present PIP-D, which employs an auxiliary decoder and two adaptive strategies to learn and predict these tailored masks, potentially enhancing performance while significantly reducing computational costs during training. To verify our PIP designs, we conduct extensive experiments on the highly challenging Traveling Salesman Problem with Time Window (TSPTW), and TSP with Draft Limit (TSPDL) variants under different constraint hardness levels. Notably, our PIP is generic to boost many neural methods, and exhibits both a significant reduction in infeasible rate and a substantial improvement in solution quality.

Collaboration! Towards Robust Neural Methods for Routing Problems

Despite enjoying desirable efficiency and reduced reliance on domain expertise, existing neural methods for vehicle routing problems (VRPs) suffer from severe robustness issues -- their performance significantly deteriorates on clean instances with crafted perturbations. To enhance robustness, we propose an ensemble-based Collaborative Neural Framework (CNF) w.r.t. the defense of neural VRP methods, which is crucial yet underexplored in the literature. Given a neural VRP method, we adversarially train multiple models in a collaborative manner to synergistically promote robustness against attacks, while boosting standard generalization on clean instances. A neural router is designed to adeptly distribute training instances among models, enhancing overall load balancing and collaborative efficacy. Extensive experiments verify the effectiveness and versatility of CNF in defending against various attacks across different neural VRP methods. Notably, our approach also achieves impressive out-of-distribution generalization on benchmark instances.

Hierarchical Neural Constructive Solver for Real-world TSP Scenarios

Existing neural constructive solvers for routing problems have predominantly employed transformer architectures, conceptualizing the route construction as a set-to-sequence learning task. However, their efficacy has primarily been demonstrated on entirely random problem instances that inadequately capture real-world scenarios. In this paper, we introduce realistic Traveling Salesman Problem (TSP) scenarios relevant to industrial settings and derive the following insights: (1) The optimal next node (or city) to visit often lies within proximity to the current node, suggesting the potential benefits of biasing choices based on current locations. (2) Effectively solving the TSP requires robust tracking of unvisited nodes and warrants succinct grouping strategies. Building upon these insights, we propose integrating a learnable choice layer inspired by Hypernetworks to prioritize choices based on the current location, and a learnable approximate clustering algorithm inspired by the Expectation-Maximization algorithm to facilitate grouping the unvisited cities. Together, these two contributions form a hierarchical approach towards solving the realistic TSP by considering both immediate local neighbourhoods and learning an intermediate set of node representations. Our hierarchical approach yields superior performance compared to both classical and recent transformer models, showcasing the efficacy of the key designs.

Adaptive Stabilization Based on Machine Learning for Column Generation

Column generation (CG) is a well-established method for solving large-scale linear programs. It involves iteratively optimizing a subproblem containing a subset of columns and using its dual solution to generate new columns with negative reduced costs. This process continues until the dual values converge to the optimal dual solution to the original problem. A natural phenomenon in CG is the heavy oscillation of the dual values during iterations, which can lead to a substantial slowdown in the convergence rate. Stabilization techniques are devised to accelerate the convergence of dual values by using information beyond the state of the current subproblem. However, there remains a significant gap in obtaining more accurate dual values at an earlier stage. To further narrow this gap, this paper introduces a novel approach consisting of 1) a machine learning approach for accurate prediction of optimal dual solutions and 2) an adaptive stabilization technique that effectively capitalizes on accurate predictions. On the graph coloring problem, we show that our method achieves a significantly improved convergence rate compared to traditional methods.

MVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts

Learning to solve vehicle routing problems (VRPs) has garnered much attention. However, most neural solvers are only structured and trained independently on a specific problem, making them less generic and practical. In this paper, we aim to develop a unified neural solver that can cope with a range of VRP variants simultaneously. Specifically, we propose a multi-task vehicle routing solver with mixture-of-experts (MVMoE), which greatly enhances the model capacity without a proportional increase in computation. We further develop a hierarchical gating mechanism for the MVMoE, delivering a good trade-off between empirical performance and computational complexity. Experimentally, our method significantly promotes the zero-shot generalization performance on 10 unseen VRP variants, and showcases decent results on the few-shot setting and real-world benchmark instances. We further provide extensive studies on the effect of MoE configurations in solving VRPs. Surprisingly, the hierarchical gating can achieve much better out-of-distribution generalization performance. The source code is available at: https://github.com/RoyalSkye/Routing-MVMoE.

Cross-Problem Learning for Solving Vehicle Routing Problems

Existing neural heuristics often train a deep architecture from scratch for each specific vehicle routing problem (VRP), ignoring the transferable knowledge across different VRP variants. This paper proposes the cross-problem learning to assist heuristics training for different downstream VRP variants. Particularly, we modularize neural architectures for complex VRPs into 1) the backbone Transformer for tackling the travelling salesman problem (TSP), and 2) the additional lightweight modules for processing problem-specific features in complex VRPs. Accordingly, we propose to pre-train the backbone Transformer for TSP, and then apply it in the process of fine-tuning the Transformer models for each target VRP variant. On the one hand, we fully fine-tune the trained backbone Transformer and problem-specific modules simultaneously. On the other hand, we only fine-tune small adapter networks along with the modules, keeping the backbone Transformer still. Extensive experiments on typical VRPs substantiate that 1) the full fine-tuning achieves significantly better performance than the one trained from scratch, and 2) the adapter-based fine-tuning also delivers comparable performance while being notably parameter-efficient. Furthermore, we empirically demonstrate the favorable effect of our method in terms of cross-distribution application and versatility.

Deep Reinforcement Learning for Dynamic Algorithm Selection: A Proof-of-Principle Study on Differential Evolution

Evolutionary algorithms, such as Differential Evolution, excel in solving real-parameter optimization challenges. However, the effectiveness of a single algorithm varies across different problem instances, necessitating considerable efforts in algorithm selection or configuration. This paper aims to address the limitation by leveraging the complementary strengths of a group of algorithms and dynamically scheduling them throughout the optimization progress for specific problems. We propose a deep reinforcement learning-based dynamic algorithm selection framework to accomplish this task. Our approach models the dynamic algorithm selection a Markov Decision Process, training an agent in a policy gradient manner to select the most suitable algorithm according to the features observed during the optimization process. To empower the agent with the necessary information, our framework incorporates a thoughtful design of landscape and algorithmic features. Meanwhile, we employ a sophisticated deep neural network model to infer the optimal action, ensuring informed algorithm selections. Additionally, an algorithm context restoration mechanism is embedded to facilitate smooth switching among different algorithms. These mechanisms together enable our framework to seamlessly select and switch algorithms in a dynamic online fashion. Notably, the proposed framework is simple and generic, offering potential improvements across a broad spectrum of evolutionary algorithms. As a proof-of-principle study, we apply this framework to a group of Differential Evolution algorithms. The experimental results showcase the remarkable effectiveness of the proposed framework, not only enhancing the overall optimization performance but also demonstrating favorable generalization ability across different problem classes.

LLaMoCo: Instruction Tuning of Large Language Models for Optimization Code Generation

Recent research explores optimization using large language models (LLMs) by either iteratively seeking next-step solutions from LLMs or directly prompting LLMs for an optimizer. However, these approaches exhibit inherent limitations, including low operational efficiency, high sensitivity to prompt design, and a lack of domain-specific knowledge. We introduce LLaMoCo, the first instruction-tuning framework designed to adapt LLMs for solving optimization problems in a code-to-code manner. Specifically, we establish a comprehensive instruction set containing well-described problem prompts and effective optimization codes. We then develop a novel two-phase learning strategy that incorporates a contrastive learning-based warm-up procedure before the instruction-tuning phase to enhance the convergence behavior during model fine-tuning. The experiment results demonstrate that a CodeGen (350M) model fine-tuned by our LLaMoCo achieves superior optimization performance compared to GPT-4 Turbo and the other competitors across both synthetic and realistic problem sets. The fine-tuned model and the usage instructions are available at https://anonymous.4open.science/r/LLaMoCo-722A.

Learning Topological Representations with Bidirectional Graph Attention Network for Solving Job Shop Scheduling Problem

Existing learning-based methods for solving job shop scheduling problem (JSSP) usually use off-the-shelf GNN models tailored to undirected graphs and neglect the rich and meaningful topological structures of disjunctive graphs (DGs). This paper proposes the topology-aware bidirectional graph attention network (TBGAT), a novel GNN architecture based on the attention mechanism, to embed the DG for solving JSSP in a local search framework. Specifically, TBGAT embeds the DG from a forward and a backward view, respectively, where the messages are propagated by following the different topologies of the views and aggregated via graph attention. Then, we propose a novel operator based on the message-passing mechanism to calculate the forward and backward topological sorts of the DG, which are the features for characterizing the topological structures and exploited by our model. In addition, we theoretically and experimentally show that TBGAT has linear computational complexity to the number of jobs and machines, respectively, which strengthens the practical value of our method. Besides, extensive experiments on five synthetic datasets and seven classic benchmarks show that TBGAT achieves new SOTA results by outperforming a wide range of neural methods by a large margin.

ReEvo: Large Language Models as Hyper-Heuristics with Reflective Evolution

The omnipresence of NP-hard combinatorial optimization problems (COPs) compels domain experts to engage in trial-and-error heuristic design process. The long-standing endeavor of design automation has gained new momentum with the rise of large language models (LLMs). This paper introduces Language Hyper-Heuristics (LHHs), an emerging variant of Hyper-Heuristics that leverages LLMs for heuristic generation, featuring minimal manual intervention and open-ended heuristic spaces. To empower LHHs, we present Reflective Evolution (ReEvo), a generic searching framework that emulates the reflective design approach of human experts while far surpassing human capabilities with its scalable LLM inference, Internet-scale domain knowledge, and powerful evolutionary search. Evaluations across 12 COP settings show that 1) verbal reflections for evolution lead to smoother fitness landscapes, explicit inference of black-box COP settings, and better search results; 2) heuristics generated by ReEvo in minutes can outperform state-of-the-art human designs and neural solvers; 3) LHHs enable efficient algorithm design automation even when challenged with black-box COPs, demonstrating its potential for complex and novel real-world applications. Our code is available: https://github.com/ai4co/LLM-as-HH.